How Do You Superpose Energy Eigenstates?

[cragar]

Homework Statement

Im trying to understand what happened in this book
they have
$|I>=\frac{|1>}{\sqrt{2}}+\frac{|2>}{\sqrt{2}}$
and then $|II>=\frac{|1>}{\sqrt{2}}-\frac{|2>}{\sqrt{2}}$
and then they say they superimpose these equations to get

$|1>=\frac{|I>}{\sqrt{2}}+\frac{|II>}{\sqrt{2}}$

did they just add the first equations together and then renormalize
it to find the new coefficients.

 

[Jesssa]

They just added them together

|I> + |II> = 2/sqrt(2) |1> = sqrt(2) |1>

So dividing by sqrt(2) gives what you have posted.

 

[Chopin]

Yes, they're just showing how to express either basis in terms of the other. Think of it just like back in high school algebra when you were solving systems of equations, and had to add multiple equations together to eliminate variables.